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Max-Planck-Institut für demografische Forschung Max Planck Institute for Demographic Research Doberaner Strasse 114 • D-18057 Rostock • GERMANY Tel +49 (0) 3 81 20 81 - 0; Fax +49 (0) 3 81 20 81 - 202; http://www.demogr.mpg.de ' Copyright is held by the authors. Working papers of the Max Planck Institute for Demographic Research receive only limited review. Views or opinions expressed in working papers are attributable to the authors and do not necessarily reflect those of the Institute. Measuring Low Fertility: Rethinking Demographic Methods MPIDR WORKING PAPER WP 2002-001 JANUARY 2002 JosØ Antonio Ortega ([email protected]) Hans-Peter Kohler ([email protected])
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Max-Planck-Institut für demografische ForschungMax Planck Institute for Demographic ResearchDoberaner Strasse 114 · D-18057 Rostock · GERMANYTel +49 (0) 3 81 20 81 - 0; Fax +49 (0) 3 81 20 81 - 202; http://www.demogr.mpg.de

© Copyright is held by the authors.

Working papers of the Max Planck Institute for Demographic Research receive only limited review.Views or opinions expressed in working papers are attributable to the authors and do not necessarilyreflect those of the Institute.

Measuring Low Fertility:Rethinking Demographic Methods

MPIDR WORKING PAPER WP 2002-001JANUARY 2002

José Antonio Ortega ([email protected]) Hans-Peter Kohler ([email protected])

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Measuring Low Fertility:Rethinking Demographic Methods.*

José Antonio Ortega †

Hans-Peter Kohler ‡

Version: 12/21/01

1. Introduction

Low fertility is a pervasive phenomenon. All European countries currently experiencebelow-replacement fertility levels, and the proportion of the world’s population living in alow fertility context continues to increase. The purpose of this contribution is to rethinkdemographic methods for the analysis of fertility from the perspective of recent research onlow fertility and to assess data requirements for such analysis. In doing so, we will mapmany of the areas where the contributions of the late Gerard Calot were particularlyrelevant.

In a low fertility context, there are some demographic tools that become central. This isthe case with tempo effects, parity specific analysis, and the introduction of fertility lifetable measures such as period parity-progression rates. While many of these methods havebeen available at least since the 1950s1, what is new is the possibility of combining thesedifferent elements of analysis. In particular, the logic and method of tempo adjustment canbe extended to any fertility measure that is calculated from tempo adjusted age- and parity-specific fertility rates. This idea, while simple, has not yet caught on in research. It is stillcommon to find some confusion about what particular methods can or cannot do. Fertilitylife table measures can eliminate compositional effects, that is, the role of the perioddistribution of women by parity, but they do not provide a measure that is free from tempoeffects2. Fortunately, the influence of parity distribution can be separated not only

* First presented at the Euroconference "The Second Demographic Transition in Europe",Bad Herrenalb, Germany, June 2001. We have received helpful comments and suggestionsfrom Laurent Toulemon, Jan M. Hoem, Evert van Imhoff and Ron Lesthaeghe. Data used inthe examples was kindly provided by Gunnar Andersson.† Departamento de Análisis Económico: Economía Cuantitativa, Universidad Autónoma deMadrid, 28049-Madrid, Spain, Email: [email protected], www:http://www.adi.uam.es/~jaortega. Part of the research was financed by CICYT, programPB98-0075. It was completed during a stay at the MPI as guest researcher.‡ Head of Research Group on Social Dynamics and Fertility, Max Planck Institute forDemographic Research, Doberaner Str. 114, 18057 Rostock, Germany. Email:[email protected], www: http://user.demogr.mpg.de/kohler.1 Hobcraft (1996) argues that the timely use of such measures would have alloweddemographers to detect the temporary character of both the baby boom and baby bust, thusavoiding the sometimes ridiculous performance of fertility forecasting and analysis duringthis period.2 The relationship between life table measures of fertility and tempo adjustment is arecurrent topic in much of the recent debate on the subject (Nì Bhrolcháin, 1992; Rallu and(footnote continued)

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conceptually but also analytically from tempo distortions caused by changes in the timing offertility, and we can gain more insight into fertility trends from such a separation.

The concept of tempo effects is therefore the first crucial point that is addressed in thisarticle. The logic and mathematics of tempo effects are connected with the idea ofdemographic translation introduced by Ryder (1964, 1980) and developed further by Foster(1990), Calot (1993) and Keilman (1994, 2001)3. Despite the importance of tempo effects inthese analyses, their goals are different. Demographic translation is concerned with thetransformation from cohort to period measures of fertility and vice versa. In contrast, tempoadjustment does not concern itself with any cohort-period transformation. It was introducedby Bongaarts and Feeney (1998) (henceforth B-F) in order to obtain “the total fertility ratethat would have been observed in a given year had there been no change in the timing ofbirths during that year” (B-F: 275)4. It is therefore based on a counterfactual: assuming nopostponement, what would the TFR be? This requires an analysis of the effects ofpostponement and the development of measures that compensate for those effects (Ortegaand Kohler, 2002). While the procedure is used by B-F only for the adjustment of the TFR,the formula works at the age-specific level (B-F, 277). It can therefore be employed toadjust each age-specific fertility rate independently. In this sense, the B-F formula is aspecial case of tempo adjustment where the adjustment ratio is the same for all the rates.

The difference of approach between tempo adjustment and the translation approach isquite subtle, since the latter also provides a decomposition of period fertility in a tempo anda quantum index (Ryder, 1980; Hobcraft, 1996). However, the meaning attached to tempoand quantum is different. Within the translation approach, the idea is to relate the momentsof the distribution of period fertility (period TFR, mean age, variance, asymmetry, …) tothose of cohort fertility. When only the first two moments are considered (cohort TFR andcohort mean age), the component associated with the cohort TFR is the quantum component,whereas the one associated with the mean age (more precisely, with the derivative of themean age) is the tempo component. This quantum component cannot be interpreted as “thetotal fertility rate that would have been observed in a given year had there been no change inthe timing of births during that year”, as in the B-F approach; that is not its purpose. Thetiming index for a particular year is calculated as the sum across cohorts of the proportionsof completed cohort fertility that took place during the year in question5. This index will belarger (or smaller) than one if fertility is being anticipated (or postponed). As a result, if anyevent takes place after the year of reference (for instance, a war), and some cohorts see theirfertility permanently reduced, this procedure would lead to an ex-post interpretation of thisas an anticipation of fertility. This is because the proportion of fertility cumulated before thewar was larger for those cohorts than could have been expected at the time. Ward and Butz(1978) and Butz and Ward (1979) saw this problem and referred to it as an Ex Post TimingIndex. They proposed an Ex Ante Timing Index which is similar to the timing index

Toulemon, 1993b; van Imhoff and Keilman, 2000; Bongaarts and Feeney, 2000; vanImhoff, 2001). The same can be said of measures based on duration specific fertility rateswhich control duration but are also affected by tempo changes. It is possible to extend thelogic of tempo adjustment to that context as does Brass (1990), defining postponement forduration-specific rates.3 With further contributions by Pressat (1969), Leguina (1976), Deville (1977), Chavez(1979), Feichtinger (1979) and Keyfitz (1985). Personal perspectives and reviews of methodare found in Hobcraft et al. (1982), Keilman and van Imhoff (1995), van Imhoff (2001).Hobcraft et al. refer to demographic translation as cohort-inversion.4 Lack of understanding of the difference between demographic translation and tempoadjustment is what lies behind the recent debate on the B-F procedure (B-F; Keilman andvan Imhoff, 2000; Kim and Schoen, 2000; Bongaarts and Feeney, 2000). As Keilman andvan Imhoff point out, the reason might be the discussion by B-F of the similarity betweencohort fertility and a moving average of tempo adjusted TFRs.5 As a matter of fact Ryder (1980) estimates the tempo index in two alternative ways thatlead to similar results. One is based in the translation formulas using the change in the meanage, while the other uses the sum of proportions of completed cohort fertility.

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produced by Ryder but which only uses past information in the cohort completion process.They use economic models to forecast future fertility with the caveat that these models alsorequire forecasting of the economic series themselves. This concept of tempo effectsremains different from that of B-F in any case, since the idea is still connected to atranslation concept irrelevant for the B-F counterfactual.

The practical relevance of tempo adjusted fertility measures stems from the extensivechanges in childbearing age which occur in low fertility countries. In most countries,childbearing is being postponed (Council of Europe, 2000; Frejka and Calot, 2001b; Kohler,Billari and Ortega, 2001). This is not a new phenomenon. Hooker (1898) and Yule (1906a)studied the consequences of postponement at the turn of the century, and they developed themain ideas. They proposed the use of the mean age to measure postponement6 and showedthat it was the speed of change of postponement that affected the marriage or birth rates.This analysis was connected with empirical evidence on the effects of abnormalcircumstances, such as the Prussian war, on marriage and fertility rates. Rates fall belownormal levels during those years, but there is recuperation after the war. Today we knowthat this pattern of rebound effects has been characteristic of short-term fluctuations in vitalrates both in historical and in present times (Lee, 1997; Reher and Ortega, 2000). Theeffects of postponement and recuperation were again at play during World War II and thesubsequent baby boom. Hajnal (1947) made a deep qualitative analysis of postponement inthis context. He also pointed out that the widespread use of family limitation was makingfertility postponement more relevant. He defined postponement in a general way verysimilar to the idea of tempo effects: “It is not even necessary to suppose that at the time the‘postponement’ takes place […] people have the idea clearly in their minds that they willlater have the children they are ‘postponing’” (p. 151). From Hajnal’s time until today,tempo effects have played an important role in explaining fertility trends. First, there was aprocess of fertility anticipation which was essential for an understanding of the baby-boomprocess. This is the motivation for the work of Ryder (1964, 1980), Pressat (1969) andDeville (1977). Subsequently, there has been a surge of interest in studying fertilitypostponement, which is related to the fact that a delay in childbearing has become apervasive characteristics of fertility patterns in low and lowest-low fertility countries.

Another crucial concept in the analysis of low fertility is the study of parity-specificfertility. There is broad agreement on this both with regard to cohort and period fertility(Lutz, 1989; Ní Bhrolchain, 1992; Rallu and Toulemon, 1993; Keilman, 1993; Hobcraft,1996): the reasons and contexts for having a first child are generally different from those forhaving a second or a third7. Since the proportion of births of higher orders is becoming verysmall, these three transitions at least should be studied individually. There is a further reasonto take parity into account in the context of demographic translation or tempo adjustment.As Hobcraft (1996) and B-F eloquently argue, tempo adjustments should be inferred fromtrends in parity-specific mean ages at childbearing. Otherwise a reduction in quantum mightbe taken as a tempo effect since the overall mean age generally declines when the proportionof higher order birth does. One example of this is shown by Lotka and Spiegelman (1940).Ryder (1980) was aware of this and, even within a quantum-tempo decomposition ofgeneral fertility, he devised an ad-hoc procedure to correct the timing index for the effect ofa quantum change on the mean age at childbearing. Basically his procedure requires anestimate of the mean age at birth of the first child, and an estimate of the average inter-birthinterval. This may be an interesting approach when no other information is available, butwhen it is possible to work with parity-specific births separately this is preferable, no matter

6 Their analysis of postponement is basically connected to age at marriage. Since they didnot have data on age at marriage they used the proportion of minors at marriage instead.They probably did not know that such data was already available in Ogle (1890). Hookerand Yule entered a very interesting debate about the possible causes of marriagepostponement, Hooker favoring the spread of education and changes in the desired standardof comfort and Yule economic factors such as price changes (Yule 1906a, 1906b; Hooker,1906).7 See, for instance, Namboodiri (1972), Seiver (1978), Louchard and Sagot (1984), deCooman et al. (1987) or Heckman and Walker (1990).

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whether we are studying cohort or period fertility. All the recently proposed tempo-adjustment procedures use parity-specific data. In the context of cohort fertility this is notalways the case. We believe that the attempt should be made of basing any inferences frompostponement, recuperation, etc., on parity-specific analysis.

In what follows, we will use the insights described above both to discuss and rethinkthe measurement of fertility with specific relevance to low fertility contexts. This discussionwill be structured as follows: in Section 2 we will present the basic tools for themeasurement of parity specific fertility: the childbearing rates. These can be classified aseither incidence rates or intensities depending on whether exposure is measured explicitly(intensities), or only along the age dimension, irrespective of parity (incidence rates). Thisdistinction proves particularly relevant in the context of tempo adjustment, since in Section3 we will show that inference about tempo should be based on intensities rather than onincidence rates. Otherwise compositional effects will bias the measurement. An example isgiven of how to adjust childbearing intensities based on the K-O framework. These can alsobe converted to adjusted incidence rates.

Tempo adjusted childbearing rates can then be used to describe fertility behavior.Fertility life table measures are particularly common in such descriptions. In Section 4 wewill discuss fertility tables for a parity-specific analysis and the different summary measuresavailable.

Both period and cohort perspectives are relevant in specific contexts. Section 5 willconcentrate on the description of period fertility. It is possible to summarize the effects oftempo change and the composition of the population by age and parity on the number ofbirths as a set of ratios. This way of describing fertility trends is particularly valuable, sincethe different demographic influences on fertility can be separated. In searching for anexplanation of fertility trends, one should concentrate on explaining the appropriatesummary fertility measures that are free from tempo and compositional effects. Section 6will examine the cohort approach to fertility analysis. There are many topics in demographythat require a cohort dimension. Its strength is the ability to track dynamically groups ofindividuals. This is essential for some demographic topics such as family dynamics, kinship,Easterlin effects, etc. For cohorts with completed fertility, the cohort perspective is notproblematical. It simply requires the use of fertility rates along the diagonals of the lexisdiagram. However, problems arise in cohort completion. We will show that the analysis oftempo and compositional effects is particularly useful in this context, as it allows thedecomposition of hypothesis about future fertility in two dimensions: quantum and tempo.The adjusted childbearing intensities can be used as a basis for the future evolution ofquantum. The future evolution of tempo can be projected based on the mean ages atchildbearing leading to different postponement scenarios. For each of these it is possible tocomplete fertility for the cohorts which presently are of childbearing age, in a way which isboth demographically coherent and takes into account all the information available.

2. The Basic Components: Childbearing Rates

We will start our analysis with the most basic components of common fertilitymeasures: the childbearing rates. Depending on the sophistication of the analysis, rates canbe made specific for a number of dimensions, the most common being the mother’s age orthe mother’s birth cohort, parity, marital status, and duration from marriage or last birth8.These measures are obtained by dividing the number of births occurring to mothers in aspecific category by a measure of exposure, that is, person-years lived by a certain group ofwomen. Depending on the exposure measure in the denominator, we distinguish fertilityintensities and incidence rates. If we divide by a measure of exposure in a specific category,we speak of occurrence-exposure rates or, following Hoem and Hoem (1989), childbearingintensities. These rates are also referred to as rates of the first kind or simply rates (Calot,

8 Other possibilities include the father’s characteristics, education level, race or ethnicity,place of residence, etc. (see Lutz, 1989).

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2002). When the denominator is a measure of exposure of all women in the age category,we call them incidence rates following Finnäs (1980) and Borgan and Ramlau-Hansen(1985). Lotka and Spiegelman (1940) and van Imhoff (2001) use the term frequencies forthese rates. They are also called rates of the second kind, reduced events, and, again, justrates. There is a straight relation between them. If we call mC(a) and fC(a) the childbearingintensity and incidence rate for the women of class C and age a, E(a) and EC(a) exposure toall women of age a, and to those in class C, and B(a) and BC(a) births to all women age a,and to women in class C, we have the following relationship:

)()(

)(

)(

)(

)(

)(

)(

)()( am

aE

aE

aE

aB

aE

aE

aE

aBaf C

C

C

CCCC ⋅=⋅== [1]

That is: in order to transform intensities into incidence rates we simply have to multiply theformer by the proportion of exposure contributed by women of class C. As we have argued,we are especially interested in the analysis of fertility by birth order. In this particular case,the class C refers to births of a given order i, and exposure is limited to those women whocan potentially have a birth of order i, that is, women of parity i−1.

There are two kinds of considerations which determine the choice between rates andintensities: some connected to their intrinsic properties, some to measurement issues. Wewill start with the former9.

Intensities are generally advocated on theoretical grounds because, when they includeall the relevant dimensions of fertility, they can reflect the instantaneous probability that awoman in that specific category gives birth (Hoem, 1976). However, this is only guaranteedwhen the subgroups of women are homogeneous with respect to their fertility behavior. If agroup is not homogenous, the fertility intensity is a weighted average of the intensities forthe different women where the weights are proportional to the respective intensities.

A second and important intrinsic advantage of intensities in the context of parity-specific analysis is their independence with respect to earlier childbearing behavior. Sincepast births are precisely the events that lead to transitions between parities, past fertilitylevels determine the proportion of exposure by women of parity j−1, which according to [1]is the conversion factor from intensities to incidence rates. This means that trends in fertilityincidence rates result not only from changes in fertility but also from changes in thepopulation composition by parity (Whelpton, 1946). In this sense, the interpretation oftrends in fertility intensities is easier since it is free from these compositional effects. Thisproperty is particularly important in the estimation of tempo effects as discussed extensivelyin Section 3.

On the other hand, incidence rates have the advantage of their additivity: the age-specific fertility rate is the sum of the incidence rates for the different classes. This is not thecase with intensities, since the sum of intensities for different parities does not make sense.One would need to convert them to incidence rates, using [1] or a life table distribution andadd them up. This additive property also extends to the calculation of the TFR as we willdiscuss in Section 4.

When both intensities and incidence rates are available, one can choose between themaccording to the purpose at hand. The problem is that intensities require more data: class-specific exposure is required and this is not always readily available. This measurementissue very often explains why fertility incidence rates are used. Of the two factors needed inthe calculation, births and exposure, births are the most widely available.

Vital statistics often provide a decomposition of births according to age and a numberof characteristics including birth order or education. A connected choice is what kind of rateto calculate within the Lexis diagram: age-period, age-cohort, or cohort-period. The most

9 See also Wunsch (2001), Vallin and Caselli (2001), Van Imhoff (2001), Toulemon (2001)for discussions of these issues.

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common combinations in fertility studies are age-period and cohort-period. Not all countriesprovide the triple age-cohort-period classification (the Lexis diagram triangles) in their vitalstatistics, so this is not always a matter of choice. Where available, cohort-periodparallelograms may be preferable since a unique cohort can be followed making them moreapt for projection purposes10. The cross-classification of births is less of a problem when themicro data regarding births is available. This is becoming more common. It allows theresearcher to cross-classify births as required. The only limitation is that the number ofsubclasses grows exponentially with the number of dimensions, which is the so-called“dimensionality curse”.

Regarding exposure, E(a) is not generally known but it can be estimated. The mostcommon estimates are mid-year populations or a population half-count (Wunsch, 2001b).These data are generally available from the intercensal population reconstructions made bystatistical agencies. Using cross-classified births and aggregate exposure, fertility incidencerates can be calculated for many populations11. The calculation of fertility intensitiesrequires the additional knowledge of exposure specific to the different subgroups. This isusually problematic since vital statistics generally do not provide all the necessary data. Inparticular, death and migration statistics are not generally cross-tabulated according to thesame criteria as births, in particular parity. In order to reconstruct the population it is usuallyassumed that mortality and migration are independent from parity12. It is then possible toreconstruct the flows since, besides migration and mortality, the inflows into the parity j andage a category are the women of age a−1 in the previous year who were of parity j and thewomen who had a birth of order j during the period. The outflows are the women of age athat had a birth of order j+1 (Calot, 2002). This can be more reliable when combined withcensus information in the reconstruction of the intercensal population13. The reconstructionof population is only the first step in the estimation of exposure. Exposure is usuallymeasured again through the mid-year population or the population half-count. In general, aswe can see, the reconstruction of exposure for subclasses is more problematic from theperspective of data quality.

While traditional measurement is based on vital statistics and population reconstruction,there are other possibilities. The ideal system may be registration as it is carried out in theScandinavian countries. This covers the entire population, making it possible to estimateboth births and exposure accurately according to the desired characteristics. A goodalternative is the use of very large retrospective surveys, say over 100.000 women14. Smallersurveys such as the FFS are not useful in this respect since the sample size is too small toestimate parity- and age-specific rates. Based on this micro data, it is possible to estimatedirectly either the intensities or the incidence rates, although it is generally the intensities orthe probabilities that are obtained. The advantage of using individual data is that it allows

10 On these issues and how to convert between the different rates see Calot (1984a). Notealso the possibility of using fertility probabilities instead of intensities. Here thedenominator would be the population at the beginning of the period.11 A common limitation is that in some countries only birth-order within marriage isknown. This limits the parity-specific analyses that can be done (Keilman, 1993).12 See Hoem (1970), Finnäs (1980) and Borgan and Ramlau-Hansen (1985) for jointstatistical modeling of fertility, migration and mortality and conversion of incidence rates orincidence rates to intensities.13 In this context it is important that censuses keep asking about children-ever-born. Insome countries this question has been withdrawn on the assumption that fertility surveys aresufficient (INE, 2001). They are not, since they do not have enough sample size to providereliable estimates of the very small proportions of women in some age and parity categories.A connected problem with intercensal reconstruction is that the results tend to be differentdepending on whether backward or forward reconstruction is used. This is connected withviolations of the independence assumption.14 Examples of such surveys are the INSEE Enquête Famille in France (Rallu andToulemon, 1993b), the Encuesta Sociodemográfica in Spain (Requena 1997), China’s oneper thousand fertility survey (Feeney and Yu, 1987) or Russia’s 5% micro census (Scherbovand van Vianen, 2001).

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the researcher to choose the relevant dimensions with sample size being the only limitation.An example of this is the estimation of age, parity and duration-specific rates as in Rallu andToulemon (1993a), where the number of categories grows to several thousand. In some ofthese categories, exposure is very limited, leading to rates with large variance. Thealternative to the calculation of separate rates for each group is the use of a hazardregression model where the intensity is modeled as a function of a vector of characteristics(Hobcraft and Casterline, 1983; Hoem and Hoem, 1989; Lutz, 1989; Andersson, 1999,2000). This eliminates the degrees of freedom problem at the cost of introducing a modelthat assumes some kind of additive effects of the covariates.

3. Tempo Effects on Incidence rates and Intensities

Fertility trends are the result of childbearing at the individual level. Here, twodimensions are relevant: how many children are given birth to, and at what age. At theaggregate level these two dimensions are intertwined: when there are shifts in the age atwhich childbearing takes place, the date at which childbirth occurs shifts as well. Thismeans that the number of births taking place in a given period when there are age shifts inthe fertility schedule is different from the number of births that would have occurred in theabsence of an age shift. This is the basic idea behind tempo effects. Tempo effects aredefined as the proportion by which fertility changes in the presence of age shifts. We areinterested in tempo effects because, on the basis of aggregate data, they allow us to separatethe two dimensions that at the individual level are evidently different: how many and when.

The history of tempo effects has been relatively short: Bongaarts and Feeney introducedthe concept in 1998. They estimated these effects from the change in the mean ages atchildbearing for different parities in successive years. They proposed the use of Ryder’stranslation formula for a linear case, which applies a factor 1/(1−r) to the period TFR. InB-F reinterpretation, r is the pace at which the period parity-specific mean age atchildbearing is shifted15. While a useful approach, and a useful formula as a firstapproximation, some of its problems have led to reformulations.

A first problem lies in the assumption that age shifts are equal for all ages. This meansthat the B-F formula is valid only for parallel shifts in the fertility schedule. This is notgenerally the case. While the practical consequences of deviations from this pattern withregard to TFR adjustment may not be very important in most cases (Yi and Land, 2000)there is no guarantee for this. More importantly, when age shifts are not parallel the errors atdifferent ages are partially cancelled out in the overall TFR, but the procedure is inadequatefor adjusting each of the age-specific fertility rates separately. It is therefore important todevelop adjustment formulas that allow for more general shifts in the fertility schedule.Kohler and Philipov (2001, henceforth K-P) have done this for a fairly general family ofshifts, and have developed a procedure for adjusting rates in the case of variance changes.The resulting formula can be seen as a generalization of the B-F procedure, where each oneof the age- and parity-specific fertility incidence rates, fj(a), is adjusted using an age- andparity-specific tempo effect, rj(a). The adjusted fertility rates are thus given by:

fj’(a) = fj(a) / [1−rj(a)] [2]

Their article provides the formula linking the tempo effects, rj(a), to the change in the meanage at childbearing, γ, and the proportional change in the standard deviation, δ. They alsopropose an iterative procedure to estimate the overall adjusted TFR taking into accountvariance effects.

A problem common to B-F and K-P is the use of fertility incidence rates for theestimation of tempo effects. Kohler and Ortega (2001a, henceforth K-O) address this very

15 Hobcraft (1996) carried out very similar calculations in applying Ryder’s method toEngland.

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issue by using fertility intensities instead of incidence rates. In order to understand theirapproach, it is helpful to first consider the different concepts of mean and variance that canbe calculated from fertility incidence rates and intensities. It then becomes possible to seewhy inference based on incidence rates is wrong, and this in turn enables us to see how toestimate tempo effects which are free of compositional biases. The basic idea is to measuretempo effects only from the fertility behavior for the parity of interest.

What is usually referred to as the mean age at birth and variance for a particular birth-order is based on the fertility incidence rate schedule. If we are using cohort-periodincidence rates and age attained during the year the expressions for the mean, µF

C, andvariance, VarF

C, are respectively16:

( )∑

∑∑

⋅−=

⋅=

aC

aC

FC

FC

aC

aC

FC

af

afaVar

af

afa

)(

)(

)(

)(

µ

[3]

We have already shown that fertility incidence rates at a given parity are the result ofthe combination of present fertility (as measured by intensities) and past fertility (as presentin the parity composition of the population). Because of this, we cannot base our estimate ofchanging fertility behavior on the incidence rates. Two examples might clarify the kind ofcompositional effects that are present in mean ages calculated from incidence rates. First, letus consider that in previous years there has been a delay in first births combined with areduction in quantum. This is a common scenario in many countries. Let us assume thatfrom a base year onwards no further changes occur. The result is that since many womenhad their first births when the rates were higher and births took place earlier, the proportionof women at parity zero at older ages is out of equilibrium with relatively few women in thatcategory. As we move towards the future, even if there is no further change in the birthintensities, the proportions of older women at parity zero will increase and there will be ashift in the mean age at birth for parity one and above. This would seem to indicate a tempoeffect where there is none. Take a second example: let us assume that the quantum of first-birth falls dramatically from one year to the next, while higher parities continue unaltered.This will lead to fewer young women entering parity one. Over the following years, themean age at second birth will increase, but this is not due to a change in behavior. Again thiswould be taken within B-F adjustment as a tempo effect in second births. Similar argumentscould be drawn up for the calculation of variances.

A second commonly used mean age (and variance) comes from the equilibriumdistribution of women by parity. We will call this the mean age at birth of the stabledistribution, µS. This may be seen as a special application of the previous formula where theparity composition of the population instead of the current one is assumed to be theequilibrium one. Still, the dependence of the equilibrium distribution on the intensities at allparities is undesirable. Take for instance a postponement of first birth with no change inquantum. The next year the equilibrium population distribution will have relatively feweryoung women at high parities. This means that the mean age at second and higher orderbirths in the stationary distribution will increase even if there has been no change inbehavior at those parities. A possible solution would be the use, for the purpose ofestimating tempo effects, of the stationary distribution corresponding to the first year (or thesecond), to estimate the mean ages at birth in both years. While this would solve theproblem of a changing parity distribution, it is still not entirely appropriate, since the tempo

16 When using age-period or age-cohort incidence rates we have to add 0.5 to the mean age.When using age at the beginning of the year we have to add 1.

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effects we want to remove are present in the intensities. Therefore the stable distributionassociated with the unadjusted intensities is different from the stable distribution associatedwith the tempo adjusted rates. While a possible iterative system could be worked out tosolve this conundrum of jointly adjusting the rates and estimating the mean ages, a simplersolution is proposed by K-O: to use the mean age at birth calculated from the intensityschedule. The corresponding formulas are the same as [3] where the incidence rates, fC(a),are replaced by the intensities, mC(a). We call these expressions the mean age at birth andthe variance of the intensity schedule (µI and VarI). They correspond to the mean age andvariance that would be observed in a population where the distribution of women accordingto parity is perfectly uniform. This is obviously not realistic, but it serves the purpose ofestimating the tempo effects. A surprising result of the application of this mean age is thatthe usual sequence in mean ages according to birth-order is generally absent: it is possible,for instance, for the mean age at third birth to be lower than the mean age at second birth.This indicates that the profile of birth probabilities might be as young for third births as forsecond ones. While this choice is natural given the use of fertility intensities, as analternative it is possible to use a different fixed distribution to calculate mean ages andvariances.

A second aspect in the estimation of tempo effects is the presence of variance effects.Again, if the shifts of the fertility schedule are parallel, one could use the shifts in the meanage as the estimate of tempo effects. When the shifts are not parallel and variance changesas in the K-P model, the mean age and the variance computed from the observed scheduleare affected by tempo distortions. An iterative procedure is needed to derive the mean ageand variance that would have been observed in the absence of tempo effects. The procedureis essentially the same as K-P with the exception that K-O apply it to the intensity schedule.Once tempo and variance changes are estimated, the adjustment formulas are given by:

)()(

)(1

)()(’

jjjj

j

jj

aaar

ar

amam

−+=

−=

δγ[4]

Full Line for the Adjusted Schedules. Dotted: Observed.

Inte

nsity

15 20 25 30 35 40 45

0.0

0.05

0.10

0.15

0.20

0.25

Fig. 1: Adjusted and Unadjusted Intensity Schedules. Sweden 1998.

parity 0

parity 1

parity 2

parity 3+

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Where γj is the overall tempo change (the increase in the mean age of the adjusted intensity

schedule), δj is the increase in the log of the standard deviation, and ja is the mean age of

the adjusted schedule. As we can see, older ages are more strongly adjusted when thevariance is increasing and vice versa. The iterative procedure designed by K-O consists ofusing the adjustment formulas [4] based on the observed intensity schedule using asmoothing procedure to estimate γ and δ from time series of mean ages and standarddeviations. From the adjusted schedules, new time series of γ, δ and a are obtained. Theprocess continues until convergence is reached.

As an example of the adjustment formulas we can take the particular case of Sweden in199817. Table 1 shows the observed fertility intensities for births of order 1, 2, 3 and theincidence rates for births of order 4 and above. The analysis in K-O provides the estimatesfor gamma and delta as given in the last rows of the table. The a , also from K-O, are themean ages of the adjusted intensity schedule. The application of the adjustment formulas [4]leads to the particular age- and parity-specific tempo effects in Table 1 from which we canthen obtain the adjusted intensity schedules in the last columns of Table 1. The table alsoprovides the sums (the cumulated intensity) and the mean ages of the intensity schedules.Figure 1 plots the schedules. The cumulated intensity corresponds to the quantum indexused by K-O. We see that the adjustment is particularly intense for parity zero andespecially parity one. Variance effects are stronger for parities one and three and above. Forthose parities the range of tempo effects is therefore wider. Note that the tempo effects canbe interpreted as the percentage by which each observed intensity rate must be adjusted toremove tempo distortions.

4. Life Table Measures of Fertility

Life table measures are probably the central tool of demographic analysis. Most of thecommonly used measures in demography such as life expectancy, total fertility rate, parityprogression ratios, net reproduction ratios, can be interpreted as life table measures. Periodlife table measures are synthetic measures, since they do not refer to a real cohort but to asynthetic one that experienced the period rates over the lifetimes (Vallin and Caselli, 2001).Life table measures thus provide a unifying framework for the study of both period andcohort demographic indicators. In this section, we will briefly review the different life tablemethods available for the study of fertility through parity and age18. We will call these tablesfertility tables.

Like Feeney and Yu (1987), we can distinguish two kinds of fertility tables: additiveand multiplicative. Additive tables are based on age-specific fertility incidence rates. Theywere first used by Böckh (1899) in the Berlin Statistical Yearbook19. They are called

17 The original data come from Andersson (2001). They have been transformed fromperiod-cohort form to age-period by means of smoothing splines.18 There are alternative fertility life tables which use other relevant dimensions such as birthinterval and parity (Feeney, 1983; Feeney and Yu, 1987; Nì Bhrolcháin, 1987), birthinterval, parity and age jointly (Rallu and Toulemon, 1993a), age and marital status (Farr,1880, Ansell, 1874), parity only (Chiang and van der Berg, 1982; Lutz, 1989), age, parityand marital status (Whelpton, 1946; Oechsli, 1975). Also, we are dealing with pure fertilitytables, while many early examples of fertility life tables were combined with mortality,leading to reproduction tables. Systematic work on these tables stems from Kuczynski(1928, 1932), although earlier examples can be found in Farr (1880), or Böckh (1886 andother years). See Stolnitz and Ryder (1949) and Lewes (1984) for a survey and a historicalnote.19 For a number of years Richard Böckh tabulated births according to parity and age but didnot standardize the figures. In 1899 he standardized them using the age distribution ofwomen thereby obtaining fertility incidence rates for the first time. Kuczynski, who was astudent of Böckh at the Berlin statistical office, popularized the measures in England and theUnited States (Kuczynski, 1928, 1932; an example of the impact of his work is Glass et al.,(footnote continued)

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additive because the total fertility measure, the parity-specific TFR, is arrived at by addingup fertility rates for a given parity. However, this implies treating parity-specific births as ifthey were repeatable events, which they are not (Henry, 1972; Keilman 1994). Adding theparity-specific TFRs, the general TFR is obtained. This also applies when other dimensionsbesides parity are considered. In general the class-specific TFR, TFRC is given by:

∑=a

CC afTFR )(

The overall TFR is then obtained either from the addition of the different class-specificTFRs, or from the addition of the overall fertility incidence rates:

∑∑ ==aC

C afTFRTFR )(

This simplicity is both the strength and the weakness of additive life table measures.Since the TFR is merely a sum of rates, it can be regarded as a simple summary measure offertility that takes into account fertility at all ages with equal weight. This simplicity makesperiod TFR time series quite volatile, which is a strong point of the measure20. A secondstrong point is its relatively lax data requirements. Its first weak point is the lack of built-indemographic logic. As we have seen, incidence rates do not reflect the risk of giving birthfor any particular women. They are influenced by the parity distribution of women at eachage. When these tables are applied to a cohort, they acquire some demographic consistency,since they track the same group of women – or men – over time. Applied to period data, thisconsistency is lacking, and it is not unusual to find first-birth period specific TFRs of higherthan one (Feeney and Yu, 1987; Hobcraft, Menken and Preston, 1982; Keilman, 1994)21.This is not a problem if one interprets it as a sign of a very favorable – and inherentlyunstable – parity distribution together with high fertility for that order rather than just as animpossible fertility quantum. It is necessary to be aware of which application canlegitimately be used for each measure. An example of an additive table is given in Table 2where use is made of the tempo adjusted fertility incidence rates for Sweden, using the K-Oprocedure and the conversion formula [1]. We see that the tempo adjusted parity-specificTFRs are considerably larger than the observed ones for parities zero and one. This isparticularly so for the first birth, where the observed rates would suggest a childlessness ofalmost 40% as compared with 20% from the adjusted rates. Allowing for the differentparities, then, the overall adjusted TFR of 1.67 compares to the observed 1.43.22 We can alsosee the mean ages for the incidence rate schedule. As we have commented in the previoussection, these are higher with parity. There are no great differences in this case between themean ages for the adjusted and observed schedule. It is also possible to estimate an impliedparity distribution from the difference between the TFRs for different parities (Charles,

1938), and we owe to him many of the terms used in fertility analysis like total fertility.However, he neglected the analysis of fertility by parity. On the other hand, when Charles(1937) consulted him during his exile in England regarding the study of fertility by birthorder, he suggested the use of fertility incidence rates in what was to be the comeback ofadditive fertility tables which would be followed by Lotka and Spiegelman (1940). AfterWW2 Henry and Pressat took the distinction between rates of the first and the second kind.20 The volatility of the period TFR is desirable in that it makes it, at least potentially, easierto find the determinants of change. See Ní Bhrolchain (1992) for a similar position. Ryder(1980, 1986) instead, sees this as an inconvenient.21 This is also a possibility in cohort parity-specific life tables when there is parity selectivemortality or migration. Let us assume for instance that, for some reason, there is a one-timevery high emmigration of childless women in a particular country. Cohort second-orderTFR might then well be higher than one since the population composition by parity hasexperienced a change with greater weight attached to parity-one women, who are the onescontributing to parity two births.22 This figure is somewhat below the published figure of 1.50 since certain categories ofbirth have been excluded, like those born to foreign women (see Andersson, 1999).

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1937). This measure should be avoided when based on period data since tempo distortionsand parity composition can lead to very erroneous estimates.

Multiplicative life tables treat births of a particular order as a non-repeatable event.Since this is precisely what they are, this brings some demographic consistency into the lifetable calculations. They are based on fertility intensities. Having been first used by Quensel(1939) and Whelpton (1946), they were revived in their parity and age form by Park (1976),Lutz (1989), Rallu and Toulemon (1993a), Giorgi (1993) and de Simoni (1995). In theirclassic form, they assume that the intensities are piecewise constant within the respectiveage intervals. However, it is also possible to disregard this assumption and work incontinuous time. We refer the reader to K-O for such a context. The name of the tablesstems from the multiplicative nature of the intensities. The basic measure we will use is theproportion of women of parity j at age a0 experiencing at least n additional births betweenexact ages a0 and a1: nPj(a0,a1). These proportions are particularly simple to calculate for oneadditional birth:

−−=−−= ∑∏

=

=

11

101

1

0

1

0

)(exp1)](exp[1),(a

aaj

a

aajj amamaaP [5]

where we see the multiplicative nature of the intensities. When the proportion is calculatedin the remaining lifetime, we have Park’s (1976) lifetime probability of n additional births,nPj(a0). The argument a0 can also be dropped when the calculation is carried out through thewhole reproductive age span. K-O very often use the measure 1Pj(a), which they callconditional parity progression probability at age a. As we have mentioned, it is generally thecase that quantum measures based on multiplicative life tables are more stable than thosebased on additive measures. This is so because the additive measures only reflect the sum ofthe rates: when there are very high but transitory rates at several ages, they are simply addedup. By contrast, the weight given to a particular age in a multiplicative quantum measuredepends negatively on the quantum itself. For instance, the first derivative of the parityprogression ratio for parity 0 is given by:

( ) 0111

01 1)(exp)(

PamaI

P −=−−=∂∂ ∑

Age at parity j

Rev

erse

Cum

ulat

ed In

tens

ities

15 20 25 30 35 40 45

01

23

.1

.3

.4

.5

.6

.7

.8

.9

.95

.98

parity 0

parity 1

parity 2

Fig. 2: Reverse Cumulated Intensities and Prob. of an Additional Birth

Based on Tempo Adjusted Intensities for Sweden, 1998

Life

time

Pro

babi

litie

s of

an

addi

tiona

l birt

h

Page 14: fecuncidad[1]

13

That is: the derivative is equal to the proportion of childless women in the table. We seethen how the effect of a single intensity is muted according to how close the quantummeasure is to the maximum. This means, for instance, that whenever fertility is very high atyoung ages, the weight given to older ages diminishes. This is a strong point when usingcohort data, but it may be a disadvantage when we are only interested in a pure periodmeasure. In such a case it is possibly more interesting to use cumulated intensities which areadditive and can be converted in probabilities through a direct transformation. Figure 2 is auseful graphical representation of the reversed cumulated intensities. As suggested byToulemon (2001b), a double-scale graph of the cumulated intensities in the second scaleshows the lifetime probability measure, 1Pj(a).

Fertility tables are useful for the organization of the fertility intensities and thecomputation of summary measures. They are increment-decrement tables where the statevariable is parity, and access to each parity requires previous transition through the lowerstates. In order to build the tables, a first step is to calculate the transition probabilities fromthe intensities. This is a relatively tricky issue. On the one hand, there is the possibility ofusing the formulas based on Markov chain theory as in Hoem and Jensen (1982),alternatively, standard simplifications such as those described by Schoen (1988) or Palloni(2001) may be used. The problem is that these formulas do not easily capture the particularcontext of the birth sequence. The simplest approach is probably the direct estimation ofbirth probabilities instead of intensities as in Rallu and Toulemon (1993b), or the use of thesimple exponential formula which is then compatible with [5]23. In such a case we would getthe age- and parity- specific probability of birth, qj(a) as:

qj(a) = 1− exp[−mj(a)] [6]

We can then use the birth probabilities to work out the remaining life table measures. Forthe general fertility table, the remaining columns are the number of women of parity j andexact age a, Dj(a), and the number of births occurring to those women at age a, bj(a). Thesequantities are calculated iteratively over age from the formulas:

bj(a) = Dj(a) qj(a) [7]

Dj(a+1) = Dj(a) − bj(a) + bj−1(a) [8]

It is common that the last parity category, J, includes parities J and above, the rates beingfertility incidence rates for this group. The formulas are then slightly different for this lastparity:

bJ(a) = DJ(a) fJ(a)

DJ(a+1) = DJ(a) + bJ−1(a)

Note also the initial conditions:

D0(α) = N ; Dj(α) = 0, j > 0

where N is the radix of the table which is equal to the size of the synthetic cohort. Table 3shows the general fertility table based on the tempo adjusted intensities for Sweden in 1998.

One of the advantages of the fertility table is that many summary measures can beconstructed directly from the table births. For instance, mean numbers of births for womenin the synthetic cohort can be defined by rectangular sums of births in the table (de Simoni,1995) as:

23 An interesting alternative is to apply the exponential formula to six month age intervalsobtained by duplication of the rates. This would allow for two transitions within a year, butno more, in correspondence with the biological limitations.

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∑∑= =

=1

0

2

1

21)(),( 10,

a

aa

j

jjjjj abaab [9]

In particular, the completed fertility of the fertility table corresponds to b0,J(α,ω)/N =b0,J(α)/N. In our case, the value is 1.7. This is the index called PATFR by Rallu andToulemon (1993a), meaning that it is the index of total fertility obtained from a parity andage fertility table. This index is free from compositional effects (because it is based on thetable distribution of women) and from tempo distortions (because it has been computedfrom tempo adjusted intensities). The cumulated sums CF(a)=b0,J(α,a−1)/N are also a usefulmeasure since they provide the cumulated fertility before age a. They are commonly usedespecially in cohort studies, where they are referred to as incomplete cohort fertility. Fromthe sum of columns corresponding to each parity, the parity-specific total fertility indexescan be obtained, PATFRj. This can be interpreted as the proportion of women in thesynthetic cohort that had at least j+1 children. They are shown as the last row of the birthcolumn in Table 4. They also correspond to Park’s lifetime probabilities of j+1 births. In thisexample it appears that both the proportion of women having an additional birth at parity 0and 1 are very high, with less than 16% childlessness and more than two thirds of thewomen having at least 2 children. In contrast, the parity 2 index is very small, below 17%.The differences between the PATFRj and the TFRj obtained in the additive table, Table 2,are the result of the parity distribution of women in 1998 being different from that of thestable distribution. Since the values based on the multiplicative table are higher than thosefor the additive table for low parities (0 and 1) and lower for high parities (2, 3 and more),this shows that the proportion of women in the lower parities is surprisingly low. This is theresult of the roller coaster fertility phenomenon that took place in Sweden during the early1990s, when the rates for the first and second births were high for a few years (e.g., seeHoem and Hoem, 1996). Fertility beyond the third birth is almost negligible.

From the PATFRj the Parity Progression Ratios (PPRs), πj, (Henry, 1953; Ryder, 1986;Ní Bhrolchain, 1987; Feeney and Yu, 1987) can be defined. They represent the proportionof women that were ever at parity j who moved on to parity j+1. They can be calculated as

πj = PATFR j+1 / PATFR j

In our case the parity progression rates for parity 1 and 2 are 79.3% and 25.0% respectively.

The distribution of women by parity at the different ages is immediately available in thefertility table as Dj(a). If the fertility intensities remained constant long enough, these

Fig. 3: Cumulated Proportions of Women at Different Parities

Based on the Stable distribution for Sweden, 1998Age

Per

cent

age

of w

omen

at p

arity

j or

bel

ow

15 20 25 30 35 40 45

020

4060

8010

0

parity 0

parity 1

parity 2

parity 3+

Page 16: fecuncidad[1]

15

proportions would be the ones observed in the population. It is in this sense that theseproportions provide the stable age distribution of women by parity. The final distribution isparticularly evident in the last row. In this specific example the high PPRs for parities 0 and1 in connection with the low PPR for parity 2 lead to approximately half the women in thestable distribution having two children. Figure 3 shows the cumulated proportions of womenat the different parities by age.

The calculation of the additional birth proportions nPj(a,b) is more complicated for thegeneral case. The fertility table can be interpreted as the exercise of tracking the fertility of awoman initially parity 0 from age α until age ω. The additional birth proportions requiresimilar calculations, but for a woman initially parity j and age a and followed only until ageb and the birth of order j+n. They can be constructed by forming a specific fertility tablewith a radix of one chosen for the desired initial category (age a and parity j) and where theabsorbent parity is j+n. K-O call this the synthetic cohort age a and parity j. The additionalbirth proportions can then be read either as the proportion of women in the absorbent state atage b+1, or as the cumulated sum bj+n−1(a,b). Table 4 shows an example of such acalculation for women at parity 1 at thirty-five years of age. As can be seen, all theassociated measures, such as number of births or mean age, can also be applied here. In thiscase there is a 46.82% chance that the woman in question will have an additional birth, butat 2.8%, the probability of having more than two children is very low.

It is also possible to obtain birth-interval measures, since the mean birth interval fromparity j to parity j+1 is equal to the difference between the mean age at birth at parity j+1minus the mean age at birth at parity j for the women who had additional children. This canbe computed by splitting the number of births in the general fertility table into two columns:those by women that had additional children, +bj(a), and those by women that remained atparity j+1, 1bj(a). They are given respectively by:

+bj(a) = bj(a) · πj+1(a) [10]

and

1bj(a) = bj(a) · [1−πj+1(a)] [11]

One can then get the mean ages at birth for those that progressed and those that did not, andby subtracting this from the mean age at the next birth, we get the mean birth interval for thetransition for parity j to j+1 (Feichtinger, 1987). In Table 5 we show an example of thecalculations. We observe that those women who progress to second birth were muchyounger when having their first birth than those that did not (27.2 versus 33 years). Thesame applies for progression to third birth. Because of this, the difference between theoverall mean ages at birth would be an underestimate of the birth interval. The mean birthintervals obtained are therefore 3.7 years from first to second birth and 4.7 from second tothird.

It is also possible to obtain a mean birth order for different ages (de Simoni, 1995). Fora particular age a, the mean birth order would be:

=

=

+=

J

jj

J

jj

ab

abj

aO

0

0

)(

)()1(

)(

We can base any of the standard life table functions for increment-decrement tables onfertility tables. In particular, one can also define conditional person-years lived or waitingtimes (Palloni, 2001). Waiting times at age 15 and parity 0 can be calculated easily from thefertility table by calculating a number of women-years lived table Lj(a) = [Dj(a)+ Dj(a+1)]/2and adding up the number of years lived in each state. Table 6 shows such a calculation. Inthis particular case, we see how late childbearing implies that women pass most of their

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childbearing life at parity zero (15 years), and the proportion spent at parity one is very lowgiven the late progression to parity one and the high progression rate to parity two. Thesewaiting times can be generalized for other ages and parities. The basic idea is to base thecalculation on the table for the synthetic cohort age a and parity j. De Simoni (1995) alsoproposes a number of age-span measures based on person-years-lived at a given parity bywomen who progress to an additional birth and by women who do not, which are all basedon the table of Lj(a).

While we have concentrated on the measurement of fertility by parity and age, there arealternative schemes for the study of fertility (Lutz, 1989; see also footnote 18). Each of themleads to related summary measures. The most common alternatives are parity and durationlife tables. Here, it is possible to define mean birth intervals and parity progression rates(Henry, 1953; Feeney and Yu, 1987; Nì Bhrolchain, 1987), and an index of total fertility,Rallu and Toulemon’s (1993a) PDTFR, Parity and Duration TFR. Using large surveys, it ispossible to extend this to additional dimensions, as shown by Rallu and Toulemon’s (1993a,1993b) calculation of PADTFR: the parity, age and duration index of Total Fertility. Boththe summary measures proposed for parity and age and parity and duration schemes can beextended to the parity, age and duration scheme. Regarding the choice between parity andage versus parity and duration, there are good arguments for both. Tables with no durationloose sight of the low fertility period following birth and the conception-birth interval.Tables with no age loose sight of the fact that age conditions parity progression: it is not thesame for a woman to reach parity one at thirty, thirty-five or forty years. Indeed, itspostponement is one of the characteristic elements of present fertility, and one of its effectsis what K-O call the fertility aging effect: the lower fertility achieved by women who delaytheir fertility. This, together with the technical possibility of removing tempo effects withinthe parity and age scheme, is the main reason for our choice here.

5. Layers in Period Fertility Analysis: From Births to Fertility Behavior

Fertility table modeling as described in the preceding section is valid both for a periodand a cohort perspective. In the first case, rates are applied to a synthetic cohort; in thesecond, to a birth cohort. Period measures frequently are of interest, since they provide ameasure of fertility at a given moment in time. This is very important, since time is one ofthe main dimensions of change in fertility (Nì Bhrolchain, 1992). Understanding thesetrends is one of the main purposes of fertility analysis. Only by using period measures canwe gain insight into the effects which current events have on fertility. This is importantirrespective of whether we are looking at the effect of socioeconomic evolution on fertility,that of public policies, etc. Having a life table interpretation is then a good property for aperiod measure of fertility. The basic possibility opened up by the techniques described inthe preceding sections is the removal of compositional and tempo distortions from periodfertility measures. This is important because neither the composition of the population nortempo distortions on quantum are directly connected to behavior. They should therefore beremoved before trying to explain period fertility.

On the other hand, whenever we are interested in the consequences of period fertility, itis usually the number of births that matter (Calot, 2001a, 2001b; Toulemon, 2001a; vanImhoff, 2001): this determines the size of future generations with its impact on their labormarket conditions, housing, pension systems, education, etc. (Ryder, 1965). The basicpurpose of demographic analysis in this respect is the separation of the differentcontributing factors to the number of births. It then becomes relevant to explore how tempoeffects and population composition combine with fertility behavior in determining thenumber of births. In this section, we will concentrate on providing such a decomposition ofthe different layers in period fertility from births to fertility behavior.

The idea of removing compositional effects from period fertility is central in fertilityanalysis. It goes back at least to Newsholme and Stevenson (1906), who stated that “the

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corrected rate measure is a force, the crude rate the result of the operation of this force”24.Based on our discussion, there is a growing number of factors to remove from observedbirths until we get to fertility: age composition, parity composition, and tempo distortions.The first is the composition of the population by age, which is removed by using the TotalFertility Rate. The relationship between the TFR and the number of births has been studiedby Ryder (1964, 1980) and Calot (1984,1985). Calot uses the term mean generation size forthe factor that translates the TFR into births. Ryder (1980) calls age distribution factor thenumber that converts the TFR into the birth-rate25. Obviously, mean generation size is equalto the age distribution factor multiplied by the average population size (or mean exposure).Because of the shortcomings of the TFR as a measure of completed fertility, Calot (2001a;2001b) advocates that the period TFR should not be interpreted as a measure of fertilityquantum at all, but rather as a measure of period generation replacement: as the ratio of thenewborn generation to the generation of mothers.

A second factor in the number of births are tempo effects. Ideally we can remove tempodistortions using fertility intensities. These adjusted intensities can be converted back intofertility incidence rates in order to obtain a TFR measure which is free from tempo effects.This procedure for adjusting the TFR leads to a result which is analogous to the B-Fprocedure with the exception that the inference on tempo effects rests on a soundermethodological basis. We can thus give a mean tempo effect measure similar to B-F’s rwhere:

TFRAdj

TFRr

.1−= [12]

The interpretation, when put in percentage terms, is the percentage of births “missing”because of tempo effects. It can be defined separately for the different parities.

Finally, the effect of parity composition can be removed using the adjusted PATFR, thetotal fertility estimate from the multiplicative fertility table of tempo adjusted intensities.Since the adjusted PATFR is a pure index of fertility in the sense that it is free from tempoand compositional distortions, it is called the Period Fertility Index, PF by Ortega andKohler (2002). We can define a parity distribution effect similar to r which we call d, theparity distribution effect:

1.

. −=PF

TFRAdjd [13]

We have inverted the sign so that this index will be positive when the parity compositionfavors high fertility, and negative when it rather leads to low fertility. Again, the paritydistribution effect can be defined separately for the different parities. They could also bedefined for a particular age.

This procedure provides a coherent partition of the TFR into its demographiccomponents which can be fruitfully exploited for forecasting or analysis purposes. Puttingtogether [12] and [13], we have:

TFRt = (1−rt) · (1 + dt) · PFt [14]

24 They try to correct for age and marital composition: “the corrected birth-rate must be ameasure of fertility, which operating upon a population of given constitution as to age, sexand marriage, produces as its result the crude birth-rate” (Newsholme and Stevenson, 1906:35).25 It is possibly better to use the mean generation size since introducing the age distributionleads to confounding factors: past fertility determines the size of the population at youngages. Therefore after a baby-boom the crude birth rate might fall merely because theproportion of women at childbearing ages in the population falls (Whelpton, 1963).

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From the definition of the mean generation size, G, we can directly connect the number ofbirths to the adjusted PATFR, our measure of quantum that is free from non-behaviouraleffects, as:

Bt = Gt · (1−rt) · (1 + dt) · PFt [15]

Other combinations of terms are also possible: for instance, one can define a meangeneration size for the tempo adjusted TFR which would include Gt and rt. These issues arefurther developed in Ortega and Kohler (2002).

A minor caveat for this procedure is the sensitivity of the partition to the sequence ofoperations: the decomposition is different if the effect of parity distribution is removedbefore the tempo effect, which is also possible. We prefer the decomposition given herebecause the interpretation of the tempo effect is simpler, it refers to the actual proportion ofbirths being missed. It also provides a useful comparison to the adjustment procedures basedon incidence rates such as B-F and K-P.

In Table 7 we show examples of such decompositions for Sweden in 1990 and 1998.The first was the year when fertility reached its highest levels in Sweden for many yearsafter a very fast increase and before a very sudden collapse (Andersson, 1999). This collapsereached a trough in 1998 (Andersson, 2001). Looking at the period fertility index PF, wecan see that the peak was real: for all parities the values are much higher in 1990 than in1998. In 1990 tempo effects and parity distribution played in opposite directions: therewere some tempo effects, on average 11% of the fertility level, but the parity compositionpartially offset this effect. Tempo effects were particularly strong for first birth and theadjusted TFR is actually higher than one. Of course, this can only happen when the paritydistribution favors high fertility. The strong parity composition effect discounts that effectleading to a period fertility index of 0.89. To understand the large parity distribution effectwe have to remember that the period fertility increase happened basically at all ages. If therates had been sustained for some time, the proportion of women at higher parities wouldhave been larger, and there would had been fewer births accordingly. This is what the ratio dpicks up. The situation is different in 1998. Tempo effects are more important than before,especially for first and second birth. On the other hand, now the compositional effect is alsoleading to low fertility, as we can see from the negative values of d at parities 0 and 1. Aswe have shown in Table 3, this is precisely the result of the fertility peak of the 1990s. Sincemany women had progressed at the time to higher parities, there is a lower observed fertilityat low parities and substantially more births at high parities (d is 17.7% and 91.7% forparities 2 and 3 respectively). Table 7 also provides the PATFR so that the alternativedecomposition can be worked out as well. The mean tempo effect in PATFR is obtained bycomparing the adjusted PATFR and the unadjusted one.

It may also be of interest to look at mean ages and birth intervals. These are shown inthe table as well. We see how substantial postponement has taken place irrespective ofwhether we look at the mean age of the stable distribution or that of the incidence rateschedule. The mean birth intervals can also be estimated from the life table of observed ortempo adjusted intensities. In the latter case, we obtain a tempo adjusted birth interval. Wesee that the results in this case are not very sensitive to tempo distortions. They also operatein different directions for the transition to second and third births. Comparing the figures for1990 with those of 1998 we see that the birth intervals have widened slightly. This should beseen as a period effect: in the year of high fertility, 1990, not only was fertility higher butthe birth interval shorter. It is also possible to compute the rest of the fertility measurespresented in Section 5.

It is important to bear in mind that these are all period measures, they therefore measurefertility in a given year and should not be interpreted as what will be observed in cohorts ofwomen. That is the object of cohort analysis which we analyze in the next section.

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6. Cohort Fertility

Cohort fertility is a particularly interesting strand of fertility analysis. In its presentform, it owes much to the work of Whelpton (1949), Ryder (1951) and Hajnal (1947)26. Theidea is to follow the fertility experience of a cohort of women (or men) through time. Whenthe fertility rates are of the cohort-period or cohort-age type, cohort analysis follows fromsimply reading the rates along the diagonal of the Lexis diagram instead of the vertical lines.What lends cohort fertility its attraction is that the rates refer to roughly the same group ofwomen. If parity-selected migration and mortality are not important, there will be acorrespondence between fertility measures based on the cohort additive fertility tables andretrospective measures of fertility based on children-ever-born to women in a survey(Wunsch, 2001a). This therefore provides an interesting connection between fertility andfamily dynamics which is not so straightforward in period analysis.

The key characteristic of cohort fertility dynamics is that its variability from cohort tocohort is very small compared to the variability over time of period total fertility. There aretwo reasons for this. One is purely mathematical: since the variability of period fertility isvery high, averaging over different periods, which is what cohort fertility does, must reducethe variance. The second is connected to the concept of a cohort: women may decideaccording to period circumstances when to have children but they might have an ex-anteidea about what ultimate level of fertility they want. If the variance of this ex-ante idea isnot large across cohorts because of common socialization, the variance of cohort fertilityshould be smaller. Ryder (explained in Hobcraft et al, 1982) actually showed that thevariance of fertility is reduced to similar levels irrespective of whether we follow realcohorts or other combinations of period fertility which do not correspond to a cohort. Thisdoes not necessarily mean that cohorts have no effect. On the contrary, while there arereasons why the variance should be smaller, there are also factors leading to inter-cohortvariability like cohort size, education (which is always more comparable for people whowere educated at the same time), contraceptive knowledge, etc. (Ryder, 1965). These canlead to differences in fertility across cohorts. In recent times, interest in the study of cohortfertility has grown as exemplified by Lesthaeghe and Willems (1999), Lesthaeghe (2001),Frejka and Calot (2001a; 2001b), van Imhoff (2001), and K-O.

From an analytic point of view the study of fertility does not present particulardifficulties. The fertility measures presented in Section 4 are all applicable both to cohortand period analysis. The only difference is in the notation. The cohort equivalent of the TFRis generally called Completed Cohort Fertility, CCF. The reason for this is that it is possibleto use census information to provide estimates of cumulated fertility and parity distributionat a particular moment in time (Sallume and Notestein, 1932). While demographictranslation formulas can be used for translating period measures into cohort measures andvice versa, this should not in fact be done. Not only is it an impossible task (van Imhoff,2001), but also a quite useless one: if we know the matrix of rates over time we shouldcompute both cohort and period measures directly from the rates.

The main problem from the point of view of cohort analysis is cohort completion: whatwill the completed cohort fertility be of the generation still of childbearing age. This issue isvery much connected to fertility forecasting, and we can see cohort-completion methods asforecasting methods and vice versa. Akers (1965) draws a distinction between the periodmethod, the cohort method and the parity progression method in forecasting births orcompleting cohorts. This distinction is still valid today. The period method is based on theassumption that the period TFR remains constant or follows a specific trajectory. The cohortmethod is based on some hypothesis about the completed cohort fertility of differentcohorts, the “remaining” births to these cohorts being spread through time. The parityprogression method is based on the calculation of birth intensities that are specific to, at

26 Earlier studies of fertility using cohorts of women are Ansell (1874) and Maynard (1923).Sallume and Notestein (1932) focused on completed fertility of birth cohorts.

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least, parity and age, and possibly marital status and birth interval. It is then assumed thatthese intensities will remain constant or will follow a particular trajectory over time.

The period method of cohort completion/fertility projection is possibly the first(Whelpton, 1928) and by far the most commonly used despite its methodologicalshortcomings (Lee, 1974 , 1993; Tuljapurkar and Boe, 1999). It is based on the assumptionof future paths for the total fertility rate and the application of a generally fixed age-specificfertility schedule. It is useful because of its simplicity and relatively lax data requirements.This makes the introduction of statistical forecasting methods easier. However, it ismisleading in periods of fertility change, that is, precisely in those periods when forecastsare more relevant. Fertility peaks associated merely with fertility anticipation would betaken to be more permanent, instead of assuming that they will end when tempo stabilizes.Conversely, fertility observed in times of postponement might be interpreted as downwardtrends. The particular applications of Lee and Tuljapurkar and Boe get around this problemby introducing ad-hoc ultimate levels of fertility. Projections based on parity- and age-specific fertility incidence rates such as Yi and Land (2001) can also be considered as periodprojections. Their approach is based on assuming an ongoing change in the shape of theincidence rate schedule as given by its mean age and variance. Even if parity is considered,the use of incidence rates implies that the parity composition of the population is not.

The cohort method for cohort completion consists in setting up a “target” fertility levelfor the different cohorts and then using an appropriate method to distribute the births. Anexample of such a procedure has recently been advanced by Lesthaeghe (2001). It is basedon taking a cohort as a reference and measuring “postponement” and “recuperation” againstit. Postponement is defined as the relative lagging of cumulated fertility in a given cohortborn in year b before age 30 with respect to the reference cohort. Recuperation is given bythe relative catching up of fertility after age 30. For a given cohort, completed cohortfertility is then given by two parameters: kb, the intensity of postponement, and Rb, theintensity of recuperation:

CCFb = CCF* − dn(30)·kb+ rn(50)·Rb

dn(a) and rn(a) being some country-specific calendars of age-related relative postponementand recuperation defined over the intervals 15-30 and 30-50 respectively. This methodsolves some of the issues involved in cohort methods such as how to use the informationfrom incomplete cohorts. It is here used to estimate the kb parameters for all cohorts and theRb parameter for those cohorts that are over 30. It is then possible to extrapolate based ontime series of kb and Rb. Analogously incomplete cohort fertility can be worked out as:

CFb(a) = CF*(a) − dn(a)·kb , a ≤ 30

CFb(a) = CF*(a) − dn(30)·kb + rn(a)·Rb , a>30

The simplicity of this method makes it a useful descriptive device where the two parametershave a meaningful interpretation, but in forecasting fertility it relies on many hidden orexplicit assumptions. A first problem arises from the constant fertility trough at age 30 andthe discontinuity in modeling before and after that age. The age at which this troughhappens is a function of the decline in quantum and the difference between the mean ages ofthe fertility schedule for the reference cohort and cohort b. This will inevitably shift overtime. Where both k and R are large, the discontinuity around age 30 may lead to spikesaround age 30 that might not be very congruent. It would be interesting to analyze each birthorder separately as in Bosveld (1996). The reason is that the constants k and R when definedfor births of all orders must simultaneously play the role of both quantum and tempo. Take acase like that of Spain, where CCF has declined fast and there has been intensepostponement of first births. The model will say there is no recuperation, which means thatthere are fewer births at older ages than before. This is not surprising if fertility at higherparities has fallen and it was very high in the cohort of reference. Recuperation could onlymake sense at a parity-specific level, meaning that the first births that did not happen atyoung ages are taking place at older ages. This is the reason why the method works forcountries like the Netherlands, where quantum has not changed much in the last thirty years,but it does not work in countries with large quantum changes. A second aspect is that the

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method provides a comparison between cohorts. In order for the procedure to havepredictive content, the comparison with the reference cohort would have to contain relevantinformation for the prediction of cohorts having children now. In countries that haveexperienced sharp socioeconomic changes in the interim, like those of Eastern European,Spain, or Portugal, it is unclear whether one can gain predictive power through comparisonwith a cohort which lived under very different conditions. This last problem is not exclusiveto Lesthaeghe’s method, but also applies to cohort methods in general.

The parity progression approach consists of using period intensities specific to severaldimensions including parity. Here the idea is that the most relevant way of obtaininginformation on what current childbearing generations will do is to look at what women inthe same circumstances were doing in the last periods for which data is available. Earlyexamples are Akers’(1965) and Ryder’s (1980, 1986) completion of cohort fertility whichwas based on the last period intensities specific for parity, marital status and birth interval.

K-O’s method of cohort completion also belongs to the parity progression approach.This method is based on the projection of fertility intensities for alternative scenarios ofquantum, tempo and variance. While it can be applied to any given set ofpostponement/quantum scenarios, they concentrate on two: the postponement stops and thepostponement continues scenario. The postponement stops consists of using the currenttempo adjusted parity- and age-specific fertility intensities for completing cohort fertility.While this is an interesting scenario to consider, it is probably not the most likely.Postponement trends seem to be very persistent (Kohler, Billari and Ortega, 2001), thereforeit is more likely that postponement will continue in the future. The postponement continuesscenario assumes that the tempo-adjusting parameters, γ and δ, will continue in future andthat the quantum, given by the cumulated intensity for a given parity, will remain unaltered.It is similar to Yi and Land’s (2001) scenario but defined for tempo-adjusted fertilityintensities instead of tempo-distorted incidence rates. K-O provide formulas for obtainingcompleted cohort fertility and related measures directly from the intensities observed.Conceptually this is the same as introducing tempo and variance effects back into the futureintensities by means of system [4], the difference being that now the adjusted rates comefrom the profile of adjusted fertility schedules in the reference year T. The mean age andvariance of the adjusted schedule will change over time. In a given year t they will be givenby:

( ))(2exp)()(

)()()(22 TtTsts

TtTata

jjj

jjj

−⋅=

−⋅+=

δ

γ[16]

We can then transform the adjusted intensity schedule into one that has mean and variancesgiven by [16]. Once this adjusted schedule has been obtained, we can get the observedintensities for the given year t from system [4]. While formula [16] is valid only for thepostponement continues scenario with constant γ and δ parameters, it can be generalized tocover any future profile of tempo and variance parameters. This makes the procedureamenable to the incorporation of statistical forecasting methods for the future paths of 1pj, γj

and δj. K-O is an improvement on previous parity progression projections in the explicitconsideration of tempo. This was a flaw of earlier attempts that were based on tempo-distorted measures of fertility and did not take into consideration the effect of tempo changeon the intensities. This is the reason, for instance, why earlier attempts such as Akers’(1965) needed to combine, even in an ad-hoc manner, parity-progression with a cohortapproach to ensure sensible demographic results. Kohler and Ortega (2001b) apply themethods to three countries, Sweden, the Netherlands and Spain. A comparison of completedcohort fertility with the projected cohort fertility for different base years, shows that theprojection fares well when the assumptions about quantum, tempo and variance are close tothe period evolution of fertility as in the Netherlands, while they fail in years of unusualdevelopments like the Swedish roller coaster fertility of the 1990s. While this is animprovement on previous methods, more effort needs to be put into forecasting turningpoints both in the quantum and tempo of fertility.

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The reason why procedures based on parity progression lead to results different fromperiod methods basically lies in the feedback effects of progression at lower parities: sincesecond births only occur to women at parity one, the parity progression rate to second birthdeclines when the first birth is postponed. Such feedback effects are called fertility agingeffects by Ortega and Kohler (2002). These effects can be partially or totally compensatedwhen births at higher parities are also being postponed. Empirical patterns show that fertilityaging has a stronger effects in some countries like Spain or Sweden than in others like theNetherlands.

7. Discussion

In this article we have provided an overview of old as well as more recent methods forthe analysis of fertility. We have tried to emphasize the flexibility of the methods and thebroadness of interest of demographers and fertility researchers. It is a fortunate circumstancethat we can adapt the methods to our interests, and that no matter what our interests are,there are a small number of key issues that we should address. It does not make sense towage war over the question of whether the TFR is a good or a bad measure. It is good insome contexts, too simple in others, and indeed too complex in others. Accordingly, it is thespecific context of our research which should guide us in the adoption of the appropriatemeasures. We have tried to present a wide range of techniques that are at the disposal of theresearcher and we have placed particular emphasis on how to combine them.

In a context of low fertility a key issue is the separate analysis of fertility by birth order.Since most births are of an order lower than 3, the separate analysis of transition to first,second and third birth is sufficient. Given that in many low fertility countries a largerproportion of births occur outside marriage, it is advisable to study births irrespective ofmarital status. To make this possible, vital statistics must provide the appropriate tabulationof births, which is not always the case. We have also seen that it is important to useintensities, real exposure-occurrence rates, instead of incidence rates or rates of the secondkind. The second ones are not suitable for the study of parity progression, one of the mainpredictable factors in fertility. They also lead to better estimation of tempo effects, which isa second important predictable component of fertility. The study of tempo is also especiallyimportant in a low fertility context since the timing of childbearing becomes more flexible.The adjustment of timing to the socioeconomic circumstances can potentially lead to largevariations in period fertility. Trends in postponement may also be connected to socialinteractions. The use of tempo adjustment techniques makes it possible to ellucidate thecontribution of changing tempo to period fertility rates.

The application of life table techniques to tempo adjusted fertility intensities leads tothe isolation of the behavioral component in fertility from parity composition and tempoeffects. Only at this stage should we attempt to explain fertility trends, or make hypothesesabout future developments. We provide a toolkit of ratios for the study of period fertilitythat isolate the contribution to the number of births of generation size, mean tempo effectsand the parity distribution effect. This set of tools can be applied either to general fertility orto order-specific fertility.

The knowledge gained from the analysis of tempo-adjusted fertility intensities is alsovaluable for completing cohort fertility following the so-called parity progression method.This is based on the assumption that current patterns of childbearing according to age andparity and current postponement trends provide information about future developments.This procedure can be used either for cohort completion or fertility forecasting. It makes fulluse of those observable elements besides pure fertility quantum that have predictableimplications for fertility, such as the parity composition of the population and the existenceof tempo distortions in the presence of tempo change.

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Table 1: Observed and Adjusted Intensities. Sweden 1998. Age-Period data.Intensities x 1000 r(a,t)x100 Adjusted Intensities x 1000

Age\Parity 0 1 2 3+ 0 1 2 3+ 0 1 2 3+16 1.6 21.6 2.117 3.9 21.6 5.018 7.5 29.6 21.5 9.7 9.6 32.819 11.9 55.4 21.5 9.9 15.2 61.520 18.4 89.0 21.5 10.1 23.5 99.021 25.0 106.0 57.0 21.5 10.4 -3.2 31.8 118.2 55.222 32.4 136.8 48.9 71.9 21.5 10.6 -3.2 -6.2 41.2 153.0 47.4 67.823 40.8 164.8 58.0 61.9 21.5 10.8 -3.1 -6.7 52.0 184.8 56.3 58.024 49.0 164.2 58.4 54.4 21.5 11.0 -3.0 -7.3 62.4 184.5 56.7 50.725 63.3 183.6 55.6 60.7 21.4 11.2 -3.0 -7.8 80.5 206.8 54.0 56.326 76.1 206.7 57.6 60.6 21.4 11.4 -2.9 -8.4 96.9 233.3 56.0 55.927 87.1 209.0 52.7 62.2 21.4 11.7 -2.8 -9.0 110.8 236.6 51.2 57.128 101.6 214.8 51.5 49.9 21.4 11.9 -2.8 -9.5 129.3 243.8 50.1 45.629 112.4 217.7 52.9 48.2 21.4 12.1 -2.7 -10.1 143.0 247.6 51.5 43.830 121.0 219.6 49.0 34.5 21.4 12.3 -2.6 -10.6 153.9 250.4 47.7 31.231 118.5 219.1 46.4 33.1 21.4 12.5 -2.6 -11.2 150.6 250.5 45.3 29.832 104.8 207.1 44.6 33.9 21.3 12.7 -2.5 -11.7 133.2 237.3 43.5 30.433 92.5 180.1 40.7 31.4 21.3 12.9 -2.5 -12.3 117.6 206.9 39.7 27.934 81.6 166.7 39.9 28.1 21.3 13.2 -2.4 -12.9 103.7 192.0 39.0 24.935 75.2 149.1 36.6 23.9 21.3 13.4 -2.3 -13.4 95.5 172.1 35.8 21.136 59.3 120.3 29.8 21.7 21.3 13.6 -2.3 -14.0 75.3 139.2 29.1 19.137 44.4 92.9 22.7 18.7 21.3 13.8 -2.2 -14.5 56.4 107.7 22.2 16.338 38.8 67.1 17.4 13.7 21.3 14.0 -2.1 -15.1 49.2 78.1 17.1 11.939 29.3 48.0 12.3 10.8 21.2 14.2 -2.1 -15.6 37.1 56.0 12.1 9.340 21.0 29.9 8.7 7.8 21.2 14.5 -2.0 -16.2 26.7 35.0 8.5 6.741 12.7 18.2 4.5 4.9 21.2 14.7 -1.9 -16.7 16.2 21.3 4.4 4.242 7.1 9.8 2.7 3.3 21.2 14.9 -1.9 -17.3 9.0 11.6 2.7 2.843 4.9 5.2 1.7 2.8 21.2 15.1 -1.8 -17.9 6.2 6.1 1.6 2.344 2.6 2.8 0.4 1.4 21.2 15.3 -1.7 -18.4 3.3 3.3 0.4 1.245 0.8 1.2 0.5 1.3 21.2 15.5 -1.7 -19.0 1.0 1.4 0.5 1.1

Cum. Intensities 1.450 3.310 0.850 0.740 1.840 3.770 0.830 0.680Mean Ages 30.74 29.48 29.22 29.05 30.74 29.54 29.23 28.92Gamma x 100 21.37 11.98 -2.76 -9.46Delta x 10000 -1.44 21.57 6.44 -55.63Source: Andersson (2001) for the cohort-period observed intensities, Kohler and Ortega (2001) for

the gamma and delta parameters. Own calculations.

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Table 2: Observed and Adjusted Frequencies. Sweden 1998. Age-Period data.Frequencies x 1000 Adjusted Freq. x 1000

Age\Parity 0 1 2 3+ 0 1 2 3+16 1.62 2.0717 3.91 4.9918 7.46 0.25 9.51 0.2819 11.71 1.00 14.92 1.1120 17.73 2.90 22.59 3.2221 23.52 5.52 0.43 29.96 6.15 0.4222 29.20 10.76 0.82 0.09 37.19 12.03 0.80 0.0923 35.09 17.34 1.99 0.11 44.68 19.44 1.93 0.1124 39.66 21.49 3.09 0.35 50.49 24.15 3.00 0.3325 46.77 29.32 5.07 0.58 59.54 33.03 4.93 0.5426 50.93 37.70 7.45 1.23 64.82 42.56 7.24 1.1327 51.16 42.82 9.41 1.72 65.10 48.46 9.15 1.5828 51.52 47.24 11.86 2.19 65.55 53.60 11.54 2.0029 48.90 49.19 14.88 2.78 62.19 55.95 14.49 2.5330 44.39 48.95 16.03 2.83 56.45 55.82 15.61 2.5531 37.38 46.80 17.23 3.35 47.53 53.50 16.80 3.0232 29.07 41.33 17.77 4.21 36.95 47.36 17.33 3.7733 22.68 33.89 17.07 4.62 28.83 38.93 16.67 4.1234 18.11 29.32 17.29 4.75 23.01 33.76 16.89 4.2135 15.58 24.32 16.11 4.50 19.80 28.07 15.75 3.9736 11.72 18.89 13.17 4.42 14.88 21.86 12.88 3.8837 8.20 13.93 10.13 4.07 10.41 16.16 9.91 3.5538 6.76 9.78 7.78 3.20 8.58 11.38 7.62 2.7839 4.88 7.19 5.47 2.57 6.19 8.38 5.36 2.2340 3.36 4.44 3.86 1.92 4.26 5.20 3.78 1.6641 2.04 2.70 2.00 1.20 2.59 3.16 1.96 1.0342 1.11 1.51 1.22 0.81 1.41 1.78 1.20 0.6943 0.75 0.80 0.76 0.66 0.96 0.94 0.74 0.5644 0.39 0.43 0.19 0.34 0.50 0.51 0.19 0.2945 0.11 0.18 0.25 0.30 0.15 0.22 0.25 0.25

TFRj 0.6257 0.5500 0.2013 0.0528 0.7961 0.6270 0.1964 0.0468Mean Ages 28.08 30.38 32.69 34.42 28.07 30.43 32.71 34.33Source: Andersson (2000) for the cohort-period observed frequencies. Own calculations

as explained in text.

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Table 3: Fertility Table. Tempo Adjusted Intensities, Sweden 1998.

Women Probabilities x 1000 BirthsAge\Parity 0 1 2 3+ 0 1 2 3+ 0 1 2 3+

16 1000 0.00 0.00 0.00 2.07 0.00 0.00 0.00 2.07 0.00 0.00 0.0017 997.93 2.07 0.00 0.00 5.00 0.00 0.00 0.00 4.99 0.00 0.00 0.0018 992.95 7.05 0.00 0.00 9.56 32.28 0.00 0.00 9.49 0.23 0.00 0.0019 983.45 16.32 0.23 0.00 15.08 59.62 0.00 0.00 14.84 0.97 0.00 0.0020 968.62 30.18 1.20 0.00 23.17 94.25 0.00 0.00 22.45 2.84 0.00 0.0021 946.17 49.78 4.05 0.00 31.34 111.48 53.74 0.00 29.66 5.55 0.22 0.0022 916.52 73.89 9.38 0.22 40.36 141.89 46.29 67.76 36.99 10.48 0.43 0.0123 879.53 100.40 19.43 0.65 50.66 168.71 54.71 58.03 44.56 16.94 1.06 0.0424 834.97 128.02 35.30 1.71 60.48 168.47 55.13 50.71 50.50 21.57 1.95 0.0925 784.47 156.94 54.92 3.66 77.36 186.85 52.54 56.30 60.69 29.32 2.89 0.2126 723.79 188.31 81.36 6.55 92.33 208.11 54.48 55.89 66.83 39.19 4.43 0.3727 656.96 215.94 116.12 10.98 104.89 210.68 49.91 57.09 68.91 45.50 5.80 0.6328 588.05 239.36 155.82 16.77 121.29 216.32 48.85 45.60 71.32 51.78 7.61 0.7629 516.73 258.90 199.98 24.39 133.25 219.32 50.16 43.78 68.86 56.78 10.03 1.0730 447.87 270.98 246.73 34.42 142.62 221.53 46.61 31.22 63.87 60.03 11.50 1.0731 384.00 274.82 295.26 45.92 139.84 221.57 44.25 29.77 53.70 60.89 13.07 1.3732 330.30 267.63 343.09 58.99 124.73 211.20 42.61 30.38 41.20 56.52 14.62 1.7933 289.10 252.30 384.99 73.60 110.93 186.86 38.94 27.93 32.07 47.15 14.99 2.0634 257.03 237.23 417.15 88.60 98.53 174.69 38.21 24.92 25.32 41.44 15.94 2.2135 231.71 221.11 442.65 104.54 91.09 158.10 35.11 21.05 21.11 34.96 15.54 2.2036 210.60 207.26 462.06 120.08 72.52 129.98 28.68 19.05 15.27 26.94 13.25 2.2937 195.33 195.59 475.75 133.33 54.88 102.12 21.92 16.34 10.72 19.97 10.43 2.1838 184.61 186.34 485.29 143.76 48.01 75.10 16.92 11.88 8.86 13.99 8.21 1.7139 175.74 181.21 491.07 151.98 36.46 54.41 12.02 9.30 6.41 9.86 5.90 1.4140 169.34 177.75 495.03 157.88 26.31 34.37 8.48 6.70 4.46 6.11 4.20 1.0641 164.88 176.10 496.94 162.07 16.03 21.05 4.38 4.20 2.64 3.71 2.18 0.6842 162.24 175.04 498.47 164.25 8.91 11.49 2.68 2.83 1.45 2.01 1.34 0.4743 160.79 174.47 499.15 165.59 6.18 6.07 1.64 2.34 0.99 1.06 0.82 0.3944 159.80 174.41 499.39 166.41 3.25 3.24 0.42 1.21 0.52 0.57 0.21 0.2045 159.28 174.36 499.74 166.62 1.00 1.37 0.54 1.07 0.16 0.24 0.27 0.1846 159.12 174.28 499.71 166.88 Sum per woman 0.841 0.667 0.167 0.024

Mean Age 28.22 30.93 33.5 35.11Source: Own calculation based on intensities in table 1.

Table 4: Specific Fertility Table for a woman age 35 and parity 1

Women Probabilities x 1000 BirthsAge\Parity 1 2 3+ 1 2 3+ 1 2 3+

35 1.0000 0.0000 0.0000 158.097 35.115 21.047 0.1581 0.0000 0.000036 0.8419 0.1581 0.0000 129.980 28.680 19.050 0.1094 0.0045 0.000037 0.7325 0.2630 0.0045 102.121 21.925 16.341 0.0748 0.0058 0.000138 0.6577 0.3320 0.0103 75.103 16.921 11.880 0.0494 0.0056 0.000139 0.6083 0.3758 0.0159 54.412 12.019 9.301 0.0331 0.0045 0.000140 0.5752 0.4044 0.0204 34.366 8.476 6.700 0.0198 0.0034 0.000141 0.5554 0.4207 0.0239 21.046 4.385 4.198 0.0117 0.0018 0.000142 0.5437 0.4306 0.0257 11.491 2.681 2.833 0.0062 0.0012 0.000143 0.5375 0.4357 0.0269 6.071 1.639 2.338 0.0033 0.0007 0.000144 0.5342 0.4382 0.0276 3.245 0.421 1.212 0.0017 0.0002 0.000045 0.5325 0.4398 0.0278 1.366 0.535 1.071 0.0007 0.0002 0.000046 0.5318 0.4403 0.0280 Sum 0.4682 0.0280 0.0008

Mean Age 37.211 38.965 40.630Source: Own calculation based on intensities in table 1.

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Table 5: Mean Birth Intervals, First to Second and Second to Third Births

Births in the life tableAge \ Class 1 to 2 Remain at 1 2 to 3 Remain at 2

18 0.222 0.005 0.000 0.00019 0.950 0.023 0.000 0.00020 2.773 0.072 0.000 0.00021 5.394 0.155 0.122 0.09522 10.154 0.330 0.234 0.20023 16.316 0.621 0.548 0.51524 20.616 0.951 0.949 0.99725 27.769 1.555 1.321 1.56426 36.631 2.556 1.896 2.53627 41.748 3.748 2.289 3.50728 46.374 5.403 2.764 4.84829 49.221 7.561 3.314 6.71730 49.789 10.239 3.393 8.10831 47.551 13.343 3.404 9.66232 40.613 15.911 3.308 11.31133 30.321 16.824 2.877 12.11534 23.254 18.187 2.537 13.40435 16.368 18.588 1.954 13.58936 9.924 17.015 1.244 12.00737 5.473 14.501 0.700 9.73038 2.679 11.315 0.380 7.83239 1.240 8.619 0.176 5.72640 0.461 5.647 0.076 4.12041 0.158 3.548 0.021 2.15842 0.044 1.967 0.007 1.32943 0.011 1.048 0.002 0.81644 0.003 0.563 0.000 0.21045 0.000 0.238 0.000 0.267

Mean Ages 29.617 34.457 30.925 34.144Age next birth 33.498 35.110Birth Interval 3.881 4.185Source: Own calculation based on table 3.

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Table 6: Waiting Times at the Different Parities

Women Years LivedAge \ Parity 0 1 2 3+

16 0.999 0.001 0.000 0.00017 0.995 0.005 0.000 0.00018 0.988 0.012 0.000 0.00019 0.976 0.023 0.001 0.00020 0.957 0.040 0.003 0.00021 0.931 0.062 0.007 0.00022 0.898 0.087 0.014 0.00023 0.857 0.114 0.027 0.00124 0.810 0.142 0.045 0.00325 0.754 0.173 0.068 0.00526 0.690 0.202 0.099 0.00927 0.623 0.228 0.136 0.01428 0.552 0.249 0.178 0.02129 0.482 0.265 0.223 0.02930 0.416 0.273 0.271 0.04031 0.357 0.271 0.319 0.05232 0.310 0.260 0.364 0.06633 0.273 0.245 0.401 0.08134 0.244 0.229 0.430 0.09735 0.221 0.214 0.452 0.11236 0.203 0.201 0.469 0.12737 0.190 0.191 0.481 0.13938 0.180 0.184 0.488 0.14839 0.173 0.179 0.493 0.15540 0.167 0.177 0.496 0.16041 0.164 0.176 0.498 0.16342 0.162 0.175 0.499 0.16543 0.160 0.174 0.499 0.16644 0.160 0.174 0.500 0.16745 0.159 0.174 0.500 0.167

Waiting Time 15.052 4.901 7.960 2.086Source: Own calculation based on table 3.

Table 7: Period Contributions of Tempo, Parity Composition and Generation Size to the Number of Births.Sweden: Comparison of 1990 and 1998.

ParitiesYear 0 1 2 3 + Total 0 1 2 3 + Total

Births 1990 45182 36273 16232 4696 102383 50487 48889 47020 46321 49141 Mean Generation1998 30085 26645 9627 2462 68819 48074 48451 47805 46772 48133 Size

TFR 1990 0.895 0.742 0.345 0.101 2.083 12.4% 10.4% 10.1% 4.9% 11.0% Mean Tempo 1998 0.626 0.550 0.201 0.053 1.430 21.4% 12.3% -2.5% -12.4% 14.2% Effect ( r ) %

Adjusted 1990 1.021 0.828 0.384 0.107 2.340 14.8% 8.2% 5.1% -23.1% 8.4% Parity CompositionTFR 1998 0.796 0.627 0.196 0.047 1.666 -5.3% -5.9% 17.7% 91.7% -1.9% Effect ( d ) %Adjusted 1990 0.890 0.765 0.365 0.139 2.159PATFR 1998 0.841 0.667 0.167 0.024 1.699PATFR 1990 0.854 0.704 0.306 0.110 1.974 4.0% 8.1% 16.2% 20.6% 8.6% Mean Tempo Effect

1998 0.764 0.562 0.136 0.022 1.484 9.1% 15.8% 18.2% 10.4% 12.6% in PATFR %Mean Age 1990 26.44 29.09 31.79 33.54 28.62 26.38 29.12 31.86 34.30 28.65 Mean AgeStable Distr. 1998 28.08 30.38 32.69 34.42 29.84 28.22 30.93 33.50 35.11 29.90 Frequency Sched.Adj. Mean 1990 3.54 4.45 3.64 4.40 Mean Birth Birth Interval 1998 3.71 4.68 3.82 4.56 Interval (Unadj.)Source: Data from Andersson (2001), Own Calculations.